Preheat oven to 375°F.
Remove the skin from the chicken and discard. Tear the chicken meat from the bones, shred it, using two forks or your fingers, and reserve in a mixing bowl.
Preheat a medium-size saucepot with the EVOO over medium heat and sauté the onions till soft. Add in the garlic and spices, and toast a few seconds until fragrant. Add in the tomatoes with chilies and season with salt. Cook for 10 minutes, until the flavors are well-married and the sauce is slightly thickened. Stir in the cilantro and lime juice, and remove from heat. Add a ladleful of sauce to the bowl with the chicken and toss to combine. Reserve the remaining sauce.